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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 122034f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 122034.g3 | 122034f1 | \([1, 1, 1, -10208, -378691]\) | \(18609625/1188\) | \(7509779302212\) | \([2]\) | \(314496\) | \(1.2212\) | \(\Gamma_0(N)\)-optimal |
| 122034.g4 | 122034f2 | \([1, 1, 1, 8282, -1576843]\) | \(9938375/176418\) | \(-1115202226378482\) | \([2]\) | \(628992\) | \(1.5678\) | |
| 122034.g1 | 122034f3 | \([1, 1, 1, -148883, 21964625]\) | \(57736239625/255552\) | \(1615436969898048\) | \([2]\) | \(943488\) | \(1.7705\) | |
| 122034.g2 | 122034f4 | \([1, 1, 1, -74923, 43886369]\) | \(-7357983625/127552392\) | \(-806304977600363208\) | \([2]\) | \(1886976\) | \(2.1171\) |
Rank
sage: E.rank()
The elliptic curves in class 122034f have rank \(1\).
Complex multiplication
The elliptic curves in class 122034f do not have complex multiplication.Modular form 122034.2.a.f
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
